May said: “There is a reason for wanting to ensure that we can control migration. It is because of the impact that net migration can have on people, on access to services, on infrastructure. But, crucially, it often hits those at the lower end of the income scale hardest.”

Leaked immigration plans 'catastrophic' for industry, say employers

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The subject had been raised by Ian Blackford, the leader of the SNP’s Westminster group, who urged May to admit that immigration is “essential to the strength of the UK economy, as well as enhancing our diversity and cultural fabric”.

In reply, the prime minister argued that controls were needed in part to prevent the wages of unskilled British workers being undercut. “We continue to believe as a government that it’s important to have net migration at sustainable levels – we believe that to be the tens of thousands – because of the impact particularly it has on people at the lower end of the income scale in depressing their wages.”

Blackford accused the prime minister of “dancing to the tune of her rightwing backbenchers”.

Those comments apart, Conservative ministers were keen to avoid discussion of the embarrassing leak on Wednesday, which laid bare government thinking about one of the most controversial aspects of post-Brexit policy.

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The defence secretary, Michael Fallon, was the lone government voice, promising the government would pursue a “balanced” migration policy, when he was challenged about the paper as he toured broadcast studios launching a new shipbuilding strategy. At prime minister’s questions May was not asked directly about the leak by Jeremy Corbyn, with Labour keen to play down its own differences over migration policy.

The revelations of plans for a tough crackdown on EU migration risked reopening one of a series of fraught divisions within the cabinet over how to manage the Brexit process. May has repeatedly stressed the need to tackle migration – including during the election campaign – while the chancellor, Philip Hammond, told MPs he would have liked to see the economy placed higher up the agenda.

Downing Street rejected reports of a rift over immigration, while a senior government source stressed the leaked proposals were a draft, and had since changed several times.

May’s former cabinet colleagues – including George Osborne and the Lib Dem leader, Vince Cable, who was business secretary in the coalition government – have repeatedly identified the prime minister as the driving force behind the Conservatives’ policy of cutting immigration.

Cable said on Wednesday that May had repeatedly suppressed evidence of the benefits of foreign workers. “When I was business secretary there were up to nine studies that we looked at that took in all the academic evidence. It showed that immigration had very little impact on wages or employment. But this was suppressed by the Home Office under Theresa May, because the results were inconvenient.

“I remember it vividly. Overwhelmingly it has been the case that overseas workers have been complementary rather than competitive to British workers,” he said.

Pressed about the immigration leak by journalists, May’s spokesman insisted he would not comment, but stressed the government would be likely to phase in any new regime after a transitional period – one of the least controversial points of the leaked paper.

“I have said that free movement – the principle of free movement – will end in 2018,” he said. But he added: “We are looking at implementation in a number of areas to ensure the smoothest possible exit and to give business, and everybody else, certainty.”

Asked whether that could include changes to free movement, he said: “Immigration is one of the areas where we would be looking at an implementation period, yes.”

EU experts had warned that Brussels would be unlikely to agree to a generous transitional deal, maintaining many of the benefits of the single market and the customs union as Britain would like, if free movement comes to an abrupt end.

The Brexit secretary, David Davis, will urge MPs from all parties to back the government in laying the legal groundwork for Brexit on Thursday, as he brings the EU withdrawal bill to the Commons for its second reading.

Labour has said it will whip its MPs to vote against the bill, which it regards as a power grab by ministers. But Davis is expected to tell MPs on Thursday: “I have said at this dispatch box before, if anyone in this house finds a substantive right that is not carried forward into UK law, they should say so. No one has yet brought to my attention a right we have missed.

“We are not rejecting EU law, but embracing the work done between member states in over 40 years of membership and using that solid foundation to build on in the future, once we return to being masters of our own laws.”

The prime minister said the bill “helps deliver the outcome the British people voted for by ending the role of the EU in UK law, but it’s also the single most important step we can take to prevent a cliff edge for people and businesses, because it provides legal certainty. We’ve made time for proper parliamentary scrutiny of Brexit legislation, and I look forward to the contributions of MPs from across the house.”

May is keen to avoid an embarrassing parliamentary showdown over the bill, which will be an early test of her slim majority. At PMQs, she offered to “listen very carefully” to the concerns of MPs, after the Tory backbencher Anna Soubry urged her to consider amendments to avoid it becoming “an unprecedented and unnecessary government power grab”.